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This study looks at a real-world production planning problem that arises in a Brazilian lattice slab factory. Based on its production, which comprises a process in two stages, the main concerns of the factory are the waste of steel trusses in the first stage, setups and inventory costs in the second stage. The proposed mathematical model is based on the one-dimensional multi-period cutting stock problem with two stages and aims at the minimization of setup, production and inventory costs, subject to capacity and demand balance constraints. To solve this model, different strategies based on the column generation procedure are implemented, where two sets of subproblems are used to provide cutting patterns for each stage. Then, different rounding strategies and the relax-and-fix heuristics are applied to identify integer solutions. We based the data set on real information from a specific factory and also from specialized websites. Computational results are used to evaluate the application of the proposed model in a realistic context. [ABSTRACT FROM AUTHOR]

In this paper the cutting stock problem minimizing saw cycles is considered. In this context where objects can be cut simultaneously the trade-off between raw material and machine utilization costs is of major interest. Different processing times for each cutting pattern are considered, according to the number of items on it, different from previous works of the literature. The optimization aims to minimize both machine utilization costs due to the saw cycles and raw material usage costs. An integer mathematical formulation is proposed, and a solution method based on a column generation procedure is used. Computational tests were performed to assess the impact on costs for illustrative and real industry instances. A set of instances were also randomly generated to analyse the behaviour of the outcome of the problem in face of different technical parameters. [ABSTRACT FROM AUTHOR]

In this article, we address the Constrained Two-dimensional Guillotine Cutting Problem (C2GCP) and the Constrained Three-dimensional Guillotine Cutting Problem (C3GCP). These problems consist of cutting a rectangular two-/three-dimensional object with orthogonal guillotine cuts to produce ordered rectangular two-/three-dimensional items seeking the most valuable subset of items cut. They often appear in manufacturing settings that cut objects to produce item types of low demand, such as in the cutting of flat glass in the glass industry, rocks in the granite and marble industries and steel blocks in the metallurgical industry. To model and solve these problems, we propose a novel top-down cutting approach that leads to effective mixed integer linear programming models for the C2GCP and the C3GCP. The insight of the proposed approach is to represent the cutting pattern as a binary tree, in which the root node is the object, and branches correspond to guillotine cuts. The results of computational experiments with a general-purpose optimization solver and using three sets of benchmark instances showed that the proposed models are competitive with state-of-the-art formulations of the C2GCP and the C3GCP in quality of solution and processing times, particularly when the number of items in an optimal solution is moderate. [ABSTRACT FROM AUTHOR]

In many productive processes, two important problems arise in the production planning: the lot-sizing problem and the cutting stock problem. Generally, companies deal with these problems separately but, by considering them in an integrated way, better results can be obtained. In this paper, the integrated lot-sizing and three-dimensional cutting stock problem applied to the mattress industry is investigated, aiming at reducing costs and waste. A mathematical model of mixed integer programming was proposed and solved with an optimisation package. Computational tests based on data collected at a mattress factory were carried out, allowing the comparison of the solutions proposed by the model and the solutions adopted by the factory. Additional tests were performed with random data in order to evaluate the behaviour of the model for different cases. The results indicate that the model performs well, reducing the objective function costs for different data sets. Based on the results, some interesting options can be explored by the industry; for example, by increasing the number of cutting patterns up to a certain level, the number of possible combinations for cutting is increased, resulting in better use of the material and a consequent reduction in costs. [ABSTRACT FROM AUTHOR]

In recent years, researchers have investigated a variety of approaches to integrating lot sizing and cutting stock problems due to their high importance in the manufacturing industry. Although the mono-objective integrated problem has been considered an excellent alternative for minimising global costs, it does not include all the multiple criteria involved in the manufacturing process. Thus, to address this issue, we use a multiobjective approach and explain its importance in providing various answers to the decision maker through the Pareto-optimal solution set. We analyse existing trade-offs and correlations between each cost of the integrated problem and the related decision variables. Several computational tests are performed, which validate the efficacy of our strategy. [ABSTRACT FROM AUTHOR]

The two-dimensional level strip packing problem has received little attention from the scientific community. To the best of our knowledge, the most competitive model is the one proposed in 2004 by Lodi et al., where the items are packed by levels. In 2015, an arc flow model addressing the two-dimensional level strip cutting problem was proposed by Mrad. The literature presents some mathematical models, despite not addressing specifically the two-dimensional level strip packing problem, they are efficient and can be adapted to the problem. In this paper, we adapt two mixed integer linear programming models from the literature, rewrite the Mrad's model for the strip packing problem and add well-known valid inequalities to the model proposed by Lodi et al. Computational results were performed on instances from the literature and show that the model put forward by Lodi et al. with valid inequalities outperforms the remaining models with respect to the number of optimal solutions found. [ABSTRACT FROM AUTHOR]

This article addresses the bi-objective integer cutting stock problem in one dimension. This problem has great importance and use in various industries, including steel mills. The biobjective model considered aims to minimize the frequency of cutting patterns to meet the minimum demand for each item requested and the number of different cutting patterns to be used, being these conflicting objectives. In this study, we apply three classic methods of scalarization: weighted sum, Chebyshev metric and ɛ-Constraint. This last method is developed to obtain all of the efficient solutions. Also, we propose and test a fourth method, modifying the Chebyshev metric, without the insertion of additional variables in the formulation of the subproblems. The computational experiments with randomly generated real size instances illustrate and attest the suitability of the bi-objective model for this problem, as well as the applicability of all the proposed exact algorithms, thus showing that they are useful tools for decision makers in this area. Moreover, the modified metric method improved with respect to the performance of the classical version in the tests. [ABSTRACT FROM AUTHOR]

In the one-dimensional cutting stock problem with usable leftovers (1DCSPUL), items of the current order are cut from stock bars to minimize material cost. Here, stock bars include both standard ones bought commercially and old leftovers generated in processing previous orders, and cutting patterns often include new leftovers that are usable in processing subsequent orders. Leftovers of the same length are considered to be of the same type. The number of types of leftovers should be limited to simplify the cutting process and reduce the storage area. This paper presents an integer programming model for the 1DCSPUL with limited leftover types and describes a heuristic algorithm based on a column-generation procedure to solve it. Computational results show that the proposed approach is more effective than several published algorithms in reducing trim loss, especially when the number of types of leftovers is limited. [ABSTRACT FROM AUTHOR]

The Minimum Score Separation Problem (MSSP) is a combinatorial problem that was introduced in JORS 55 as an open problem in the paper industry arising in conjunction with the cutting stock problem. During the process of producing boxes, flat papers are prepared for folding by being scored with knives. The problem is to determine whether and how a given production pattern of boxes can be arranged such that a certain minimum distance between the knives can be kept. Introducing the concept of matching-based alternating Hamiltonian paths, this paper models the MSSP as the problem of finding an alternating Hamiltonian path on a graph that is the union of a matching and a type of graph known as a 'threshold graph'. On this basis, we find a heuristic that can quickly recognize a large percentage of feasible and infeasible instances of the MSSP. Detailed computational experiments demonstrate the efficiency of our approach. [ABSTRACT FROM AUTHOR]

Several meta-heuristic algorithms, such as evolutionary algorithms (EAs) and genetic algorithms (GAs), have been developed for solving feature selection problems due to their efficiency for searching feature subset spaces in feature selection problems. Recently, hybrid GAs have been proposed to improve the performance of conventional GAs by embedding a local search operation, or sequential forward floating search mutation, into the GA. Existing hybrid algorithms may damage individuals' genetic information obtained from genetic operations during the local improvement procedure because of a sequential process of the mutation operation and the local improvement operation. Another issue with a local search operation used in the existing hybrid algorithms is its inappropriateness for large-scale problems. Therefore, we propose a novel approach for solving large-sized feature selection problems, namely, an EA with a partial sequential forward floating search mutation (EAwPS). The proposed approach integrates a local search technique, that is, the partial sequential forward floating search mutation into an EA method. Two algorithms, EAwPS-binary representation (EAwPS-BR) for medium-sized problems and EAwPS-integer representation (EAwPS-IR) for large-sized problems, have been developed. The adaptation of a local improvement method into the EA speeds up the search and directs the search into promising solution areas. We compare the performance of the proposed algorithms with other popular meta-heuristic algorithms using the medium- and large-sized data sets. Experimental results demonstrate that the proposed EAwPS extracts better features within reasonable computational times. [ABSTRACT FROM AUTHOR]

Three-staged cutting patterns are often used in dividing large plates into small rectangular items. Vertical cuts separate the plate into segments in the first stage, horizontal cuts split each segment into strips in the second stage, and vertical cuts divide each strip into items in the third stage. A heuristic algorithm for generating constrained three-staged patterns is presented in this paper. The optimization objective is to maximize the pattern value that is the total value of the included items, while the frequency of each item type should not exceed the specified upper bound. The algorithm uses an exact procedure to generate strips and two heuristic procedures to generate segments and the pattern. The pattern-generation procedure first determines an initial solution and then uses its information to generate more segments to extend the solution space. Computational results show that the algorithm is effective in improving solution quality. [ABSTRACT FROM AUTHOR]

The research addressing two-dimensional (2D) irregular shape packing has largely focused on the strip packing variant of the problem. However, it can be argued that this is a simplification. The materials from which pieces are required to be cut will ultimately have a fixed length either due to the physical dimensions of the material or through constraints on the cutting machinery. Hence, in order to cut all the pieces, multiple sheets may be required. From this scenario arises the 2D irregular shape cutting stock problem. In this paper, we will present implementations of cutting stock approaches adapted to handle irregular shapes, including two approaches based on column generation (CG) and a sequential heuristic procedure. In many applications, setup costs can be reduced if the same pattern layout is cut from multiple sheets; hence there is a trade-off between material waste and number of patterns. Therefore, we describe the formulation and implementation of an adaptation of the CG method to control the number of different patterns. CG is a common method for the cutting stock problem; however, when the pieces are irregular the sub-problem cannot be solved optimally. Hence we implement CG and solve the sub-problem using the beam search heuristic. Further, we introduce a version of CG for instances where the number of rows is less than the number of columns. [ABSTRACT FROM AUTHOR]

The two-dimensional cutting stock problem (2DCSP) consists in the minimization of the number of plates used to cut a set of items. In industry, typically, an instance of this problem is considered at the beginning of each planning time period, what may result in solutions of poor quality, that is, excessive waste, when a set of planning periods is considered. To deal with this issue, we consider an integrated problem, in which the 2DCSP is extended from the solution in only a single production planning period to a solution in a set of production planning periods. The main difference of the approach in this work and the ones in the literature is to allow sufficiently large residual plates (leftovers) to be stored and cut in a subsequent period of the planning horizon, which may further help in the minimization of the waste. We propose two integrated integer programming models to optimize the combined two-dimensional cutting stock and lot-sizing problems, minimizing the total cost, which includes material, waste and storage costs. Two heuristics based on the industrial practice to solve the problem were also presented. Computational results for the proposed models and for the heuristics are presented and discussed. [ABSTRACT FROM AUTHOR]

This paper addresses the Ground Crew Rostering Problem with Work Patterns, an important manpower planning problem arising in the ground operations of airline companies. We present a cutting stock-based integer programming formulation of the problem and describe a powerful heuristic decomposition approach, which utilizes column generation and variable fixing, to construct efficient rosters for a six-month time horizon. The time horizon is divided into smaller blocks, where overlaps between the blocks ensure continuity. The proposed methodology is able to circumvent one step of the conventional roster construction process by generating rosters directly based on the estimated workload. We demonstrate that this approach has the additional advantage of being able to easily incorporate robustness in the roster. Computational results on real-life instances confirm the efficiency of the approach. [ABSTRACT FROM AUTHOR]

Apart from trim loss minimization, there are many other issues concerning cutting processes that arise in real production systems. One of these is related to the number of stacks that need to be opened near the cutting machines. Many researchers have worked in the last years on cutting stock problems with additional constraints on the number of open stacks. In this paper, we address a related problem: the Ordered Cutting Stock Problem (OCSP). In this case, a stack is opened for every new client's order, and it is closed only when all the items of that order are cut. The OSCP has been introduced recently in the literature. Our aim is to provide further insight into this problem. This paper describes three new integer programming formulations for solving it, and an exact algorithm based on column generation, branch-and-bound and cutting planes. We report on computational experiments on a set of random instances. The results show that good lower bounds can be computed quickly, and that optimal solutions can be found in a reasonable amount of time. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for generating the nofit polygon (NFP) that is simple, intuitive and computationally efficient. The NFP has in recent years become an important tool for handling the geometric calculations for two-dimensional irregular shape nesting problems. Its value lies in reducing the computational complexity of detecting whether two polygons overlap. The proposed NFP generator is based on the novel concept of trace line segments that are derived from the interaction of the two-component polygons. The complete set of the trace line segments contain all the boundary edges of the NFP and some internal points that need to be discarded. Algorithms for deriving the trace line segments, efficiently determining those segments that form the boundary of the NFP and the identification of holes and degenerate cases are described. [ABSTRACT FROM AUTHOR]

This paper investigates the steel plate order fulfilment problem from existing inventory, which is a generalization of one-dimensional cutting and packing problems. Semi-processed inventory of steel slabs can be burned, heated, rolled, and sheared to meet customer order requirements. This problem is an integral part of the day-to-day operations of steel plate manufacturing. A 0-1 integer programming formulation is presented and two heuristics, based on single-order formulations, are presented to quickly generate lower and upper bounds. The bounds are improved using Lagrangian relaxation and subgradient optimization, producing average optimality gaps of 2.12% on 17 real-world data sets. The upper bound improvements represent actual savings over the current solution method of 1.94%, or an average of about $230000 over a 3-week period. An economic interpretation of the Lagrange multipliers is also discussed and shown to be useful for handling rush orders. [ABSTRACT FROM AUTHOR]

This paper is concerned with the development of a customized circle packing algorithm for a manufacturer of sprockets for the motor cycle industry. Practical constraints mean that the problem differs somewhat from those tackled elsewhere in the literature. In particular, the layouts need to conform to a given structure. This is achieved by using a local search algorithm with an appropriate starting solution and a series of neighbourhoods designed to preserve the layout structure. Empirical evidence based on real data shows that the quality of the resulting solutions closely matches that of cutting patterns currently produced by human experts. Computation times average around 20-30 s per order as compared to several hours for an equivalent manual solution. [ABSTRACT FROM AUTHOR]

This paper presents a greedy randomized adaptive search procedure (GRASP) for the constrained two-dimensional non-guillotine cutting problem, the problem of cutting the rectangular pieces from a large rectangle so as to maximize the value of the pieces cut. We investigate several strategies for the constructive and improvement phases and several choices for critical search parameters. We perform extensive computational experiments with well-known instances previously reported, first to select the best alternatives and then to compare the efficiency of our algorithm with other procedures. [ABSTRACT FROM AUTHOR]

Packing optimization problems aim to seek the best way of placing a given set of rectangular cartons within a minimum volume rectangular container. Currently, packing optimization methods either have difficulty in finding a globally optimal solution or are computationally inefficient, because models involve too many 0-1 variables and because use of just a single computer. This study proposes a distributed computation method for solving a packing problem by a set of personal computers via the Internet. First, the traditional packing optimization model is converted into an equivalent model containing many fewer 0-1 variables. Then the model is decomposed into several sub-problems by dividing the objective value into many intervals. Each of these sub-problems is a linearized logarithmic program expressed as a linear mixed 0-1 problem. The whole problem is solvable and reaches a globally optimal solution. The numerical examples demonstrate that the proposed method can obtain the global optimum of a packing problem effectively. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of minimizing trim loss in cutting rectangular blanks of a single size from a rectangular sheet using orthogonal guillotine cuts. First we prove that we can obtain the unconstrained optimal layout by searching among normal multi-section layouts. Next we present an unconstrained algorithm to search for it. The unconstrained algorithm uses a branch-and-bound method with a tight upper bound. Later we discuss the algorithm for the constrained problem where the blank demand must be met exactly. Finally, the unconstrained algorithm is extended to cope with the blade length constraint. Experimental computations show that the algorithms are extremely efficient. [ABSTRACT FROM AUTHOR]

Presents a study that proposed an upper bound that can be applied to the algorithms for the unconstrained two-dimensional cutting problem. Calculation of the upper bound; Application of the upper bound to the exact cutting algorithm; Efficiency of the upper bound.

Provides information on a study which applied a cutting plane algorithm to one-dimensional cutting stock problems (CSP). Valid inequalities and dominance of the polyhedron of CSP; Evaluation of the Gomory-group of cutting planes for CSP; Description of the cutting plane algorithm combined with the column generation technique; Conclusion.

Deals with a study that proposed an efficient method for solving constrained and unconstrained two-dimensional cutting (TDC) stock problems. Background information on TDC problem; Four versions of the TDC problem that can be distinguished; Information on the General Best Strips Cutting algorithm (GBSC); Main steps of GBSC algorithms; Discussion and conclusion.

Introduces the one-dimensional bounded knapsack in the original Christofides and Whitlock top-down algorithm which combines in a nice tree search procedure Gilmore and Gomory's algorithm and a transportation routine called at each node of the tree for solving the constrained two-dimensional cutting problem. Dynamic programming properties.

Proposes a heuristic algorithm to solve an industrial application of a bidimensional cutting stock problem. Application of the algorithm to check the asymptotic behavior of the solution when a continuous stream of pieces is requested and when cutting decisions are made while orders are still arriving; Integer programming problem.

Examines a range of cutting stock problems that occur in some industries where meeting orders' due dates is more important than minimizing wastage of materials. Common cutting stock problems; Problem structuring methods appropriate for working with community clients; Computational results of the study; Conclusion to study.

Presents a cutting stock problem with two objectives. Review of related literature; Assumptions of the model; Heuristic approaches for large problems.